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## LTI Systems, Probability

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**LTI Systems, Probability**Analog and Digital Communications Autumn 2005-2006 CS477: Analog and Digital Communications**Sampling**• First consider modulation • Product with Cosine in time domain • Convolution with two impulses in frequency domain • Next consider sampling • Product with a train of impulses in time domain • Convolution with a train of impulses in the frequency domain • Nyquist sampling theorem • A bandlimited signal [-B, +B] can be characterized by its samples taken every 1/(2B) seconds. i.e., 2B samples per second • Undersampling leads to aliasing CS477: Analog and Digital Communications**LTI Systems**• Linearity If then • Time invariance • Linearity and Time invariance CS477: Analog and Digital Communications**Response of LTI Systems**Impulse response: CS477: Analog and Digital Communications**Exponentials and LTI Systems**Exponentials are eigenfunctions of LTI systems! LTI Systems can not generate new frequencies! CS477: Analog and Digital Communications**Hilbert Transformer**• A filter introducing a constant delay of 90 degrees to the input signal • Hilbert transform does not change the domain; It’s merely a convolution CS477: Analog and Digital Communications**Random Variables**• Outcomes and sample space • Random Variables • Mapping outcomes to: • Discrete numbers Discrete RVs • Real line Continuous RVs • Cumulative distribution function • One variable • Joint cdf CS477: Analog and Digital Communications**Random Variables**• Probability mass function (discrete RV) • Probability density function (cont. RV) • Joint pdf of independent RVs • Mean • Variance • Characteristic function • (IFT of pdf) CS477: Analog and Digital Communications